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What are some good undergraduate level books, particularly good introductions to (Real and Complex) Analysis, Linear Algebra, Algebra or Differential/Integral Equations (but books in any undergraduate level topic would also be much appreciated)?

EDIT: More topics (Affine, Euclidian, Hyperbolic, Descriptive & Diferential Geometry, Probability and Statistics, Numerical Mathematics, Distributions and Partial Equations, Topology, Algebraic Topology, Mathematical Logic etc)

Please post only one book per answer so that people can easily vote the books up/down and we get a nice sorted list. If possible post a link to the book itself (if it is freely available online) or to its amazon or google books page.

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closed as no longer relevant by Harry Gindi, Robin Chapman, Greg Stevenson, Harald Hanche-Olsen, Scott Morrison Jul 11 '10 at 13:29

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

This is borderline, but I think it is a legitimate question of interest to math instructors. I think its better as a community wiki though. – David Zureick-Brown Oct 16 '09 at 17:28
OK... Do I need to do something to make it a community wiki or is it already like that? – person Oct 16 '09 at 18:04
@person: Normally, to make a question or answer community wiki, you click the "community wiki" checkbox in the lower-right when you're composing or editing. But moderators have the power to convert a post to community wiki, which is what David has done here (actually, three moderators independently thought it should be converted to CW ... it's not something we do willy-nilly), so you don't need to do anything. – Anton Geraschenko Oct 16 '09 at 18:15
It's no longer possible to add useful answers to this question (as there are too many!) and it's unclear whether this question would be "allowed" by modern standards -- far too broad. As it's been popping back to the front page fairly frequently, we've decided to close it. – Scott Morrison Jul 11 '10 at 13:30
See discussion on meta:… (and remember to vote this comment up, so it is visible to others) – Victor Protsak Jul 14 '10 at 10:34

96 Answers 96

Godement "Analysis" (I,II,III,IV)

"... The content is quite classical ... [...] The treatment is less classical: precise although unpedantic (rather far from the definition-theorem-corollary-style), it contains many interesting commentaries of epistemological, pedagogical, historical and even political nature. [...] The author gives frequent interesting hints on recent developments of mathematics connected to the concepts which are introduced. The Introduction also contains comments that are very unusual in a book on mathematical analysis, going from pedagogy to critique of the French scientific-military-industrial complex, but the sequence of ideas is introduced in such a way that readers are less surprised than they might be.

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How about an anti-recommendation? Someone in another answer mentioned Steven Axler's Linear Algebra Done Right. My comment, not as someone who has used this book in a class, but as someone who has taught the students from this class during the following term: It doesn't prepare the students to use linear algebra in engineering, in physics, in chemistry, or even in branches of mathematics other than abstract algebra.

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I suppose the correct thing to ask then is: what WOULD you use? Not to be confrontational, just that the standard around these parts is pretty poor, and Axler is considered much better. Also, could you explain what they're not teaching? I'd be interested to know! Thanks! – Michael Hoffman Oct 30 '09 at 13:27
Then how about "Linear Algebra Done Wrong", available for free at – Henning Arnór Úlfarsson Nov 6 '09 at 21:47
@Gerald: I'm not sure omission of Cramer's Rule is such a big deal. Cramer's Rule is helpful for solving small linear systems (2 or 3 unknowns) since there are useful heuristics for calculating the determinants of 2x2 and 3x3 matrices, but Gaussian elimination is a more powerful and general algorithm for solving linear systems. – las3rjock Nov 7 '09 at 15:49
* Then how about "Linear Algebra Done Wrong", * Looks good, actually. Suitable for students interested in other branches of mathematics! – Gerald Edgar Dec 30 '09 at 13:35
The book is for students already familiar with matrix algebra, so this is not a valid criticism of the book. – Michael Greinecker May 25 '10 at 20:51

A Field Guide to Algebra by Antoine Chambert Loir. Covers a surprising amount of material considering the short length and minimal prerequisites

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"TOPOLOGY WITHOUT TEARS" by SIDNEY A. MORRIS is a very nice introduction in topological spaces theory, I think.

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Available online: – sdcvvc Nov 9 '09 at 14:09

Geroch, Mathematical Physics

Don't be scared by the title: it teaches algebra, topology and measure theory, using category-theoretic language.

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Indeed, that's a nice one! – Peter Arndt Mar 19 '10 at 1:11

Introduction to Topology: Pure and Applied, by Adams and Franzosa. The figures in the book are beautiful, the problems are good, and the applications are good (and unusual) to see in an undergraduate text.

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That sounds like a nice book. It's pretty expensive, unfortunately. – Ryan Budney Jan 1 '10 at 1:39

"Differential Calculus on Normed Linear Banach Spaces" by Kalyan Mukherjee.

The author is a prof at the Indian Statistical Institute (Kolkata)

This is an amazing book which introduces differential calculus in arbitrary finite dimensional spaces (thinking of the derivative as the Jacobian) as the next leap from the Apostol and Rudin level and also build in topological ideas. It then goes into manifold theory and shows how to compute tangents to curves inside lie groups. It has nice sections of things like differentiating the determinant function and the matrix multiplication function and inverse function theorem and idea of equivalence of norms.

I would strongly recommend this book to an undergrad after he/she has done the Apostol/Rudin level of calculus.

"Calculus on Manifolds" by Spivak and "Differential geometry and Lie groups" by Kumaresan are two other good books which can be read alongside it.

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I think that Linear Algebra by Friedberg, Insel, and Spence is a spectacular linear algebra book. It gets straight to the point, it provides a worked out example or two exactly when they're needed, and it has lots of interesting exercises.

There are way too many gigantic linear algebra books with colour pictures and contrived examples which often seem to obsfucate the concepts being introduced. In my mind this book is 'the mathematician's linear algebra book'; clean and concise.

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I second this choice.It balances rigorous theory and applications better then any book on the subject I've seen. Charles Curtis's old classic is also excellent and it has a much better discussion of the Jordan form and decomposition algorithm. – The Mathemagician Mar 27 '10 at 21:45

Visual Complex Analysis by Tristan Needham is awesome!

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In fact, there are others that agree with you: the book is already in the list! – Bjorn Poonen Mar 7 '10 at 7:46

For (applied) ODEs: Nonlinear dynamics and chaos by Steven Strogatz.

A very inspiring book! The explanations are crystal clear with lots of pictures. And it's funny too – the "Romeo and Juliet" illustration of 2-dimensional linear systems (Section 5.3) is a classic.

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Lebesgue Integration on Euclidean Space / Frank Jones - an extremely readable book on Lebesgue theory on $\mathbb{R}^n$ (lots of figures and geometric intuition). He constructs Lebesgue measure in a very down-to-earth manner, much more explicitly than other more abstract constructions (via Caratheodory's extension theorem or Riesz's representation theorem). In my experience, it's best to first study Lebesgue measure on $\mathbb{R}^n$ and only then point out that it's merely one instance of the general theory of measures, which is the way this book is written. It can't compare with the "tougher" books on measure theory (e.g. Big Rudin) since it doesn't discuss the Radon-Nikodym theorem and many other important theorems in measure theory, but then again the book is clearly intended for an undergraduate audience, and as for Lebesgue theory on Euclidean spaces, it provides a pretty complete picture.

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One of the best introductions to the subject out there. Should be on everyone's must-read list and great collateral reading with the more intense introductions like Big Rudin or Folland. – The Mathemagician Jul 11 '10 at 0:29

Atiyah and MacDonald, Introduction to Commutative Algebra

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Do you really think this book is undergraduate level? – Qiaochu Yuan Jul 10 '10 at 18:48
I feel that in the case of AM, this criticism misses the mark: although not everyone is an algebraist, commutative algebra is a valid subject of undergraduate study (i.e. not "too deep")! The book is very clearly written (incomparably better than van der Waerden) and is good for self-study for problem-oriented people. Nonetheless, I wish that at the time I had the courage to read Zariski and Samuel instead: in spite of being 2 volumes, it is so much more relaxed because the authors take care to $\mathit{explain}$ the material, in multiple ways, and offer both breadth and depth of perspective. – Victor Protsak Jul 11 '10 at 8:15
@Victor I totally agree on Zariski and Samuels.But if you're going to invest that much time in a tome that lengthy,then you may as well get Eisenbud.In any event,all these books will be too difficult for any but the best undergraduates,I'm sorry. – The Mathemagician Jul 11 '10 at 9:05
Right. I definitely think Reid's book is at a much more manageable level than A&M. – Qiaochu Yuan Jul 11 '10 at 9:18
The second paragraph of the introduction begins, "This book grew out of a series of lectures given to third year undergraduates ..." The authors also point out that their book is not intended as a substitute to the Zariski-Samuel or Bourbaki books. – Kiochi Jul 11 '10 at 14:33

I liked Elements of Abstract Algebra by Allan Clark, which is mainly a problem book with a moderate amount of exposition, but the problems are so well-chosen that a diligent undergraduate student working through all of them will come out with a solid knowledge of group theory, classical ring theory, and Galois theory.

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Since Numerical Mathematics has not been covered, I would recommend the following

Introduction to Numerical Analysis by Stoer et. al.

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Linear Algebra: With Applications Otto Bretscher

Used at Carleton College – nice explanations, and quite a few proofs. Presents information primarily by providing examples, definitions/axioms and then proofs.

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Taught out of it several times, didn't like it at all. Also, the smugglers used to "motivate" (hah!) linear transformations in the first edition turned into evildoers in the second. I dread to even think who they've metamorphized into now. – Victor Protsak May 24 '10 at 5:55

I had many trials and these are in my opinion the best for an introductory, undergraduate level:

ODEs: Holzner: Differential Equations for Dummies
PDEs: Farlow: Partial Differential Equations for Scientists and Engineers

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G. J. O. Jameson: Topology and Normed Spaces for an introduction to functional analysis from a topological point of view.

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lindsey childs a concrete introduction to abstract algebra

here maybe.

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It's a bit obsolete by now. – Victor Protsak May 24 '10 at 5:56

Strichartz, The Way of Analysis

Herstein, Abstract Algebra

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I also like Herstein's Abstract Algebra. – Pandora Jul 10 '10 at 21:44

Milnor's book "Dynamics in One Complex Variable: Introductory Lectures". An early version is available from his website. Suitable for advanced undergraduates, graduate students, and mathematicians.

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Most readers here will not be able to appreciate them for a simple reason, but my favorite beginners' Analysis text is that by Bröcker. No-nonsense, concise, with a slight orientation towards topology.

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Hugo Steinhaus book "Mathematical Kaleidoscope". It is kind of mathematical trivia sometimes very deep;-) It is not for learning math but for learning how to learn math in fun way.

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Complex Variables: Harmonic and Analytic Functions by Francis J. Flanigan

A nice little Dover paperback which turns the standard course on complex variables on its head. It begins by doing some multivariable calculus in the plane and harmonic functions, then proceeds to talk about complex numbers and to build analytic functions.

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I found «A (terse) introduction to linear algebra» (Katznelson) to be a much better book than Axler. It's part of the Student Mathematical Library and published by the AMS.

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Siegfried Bosch, Lineare Algebra

It's a very elegant, concise but beautifully written approach to Linear Algebra, and I love it.

Unfortunately for people who don't speak German, it has never been translated.

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I plan to post a complete reading list for undergraduates and graduate students at my blog this summer with my commentaries,but here's one I think that's available online and doesn't get nearly enough credit despite the fame of it's author: Gilbert Strang's Calculus. I wouldn't use anything else for a regular,non-honors calculus course. Carefully written,beautifully motivated with TONS of creative and SIGNIFICANT applications. I hope one day Strang finds the time to write a second edition-I have a list of improvements to suggest.

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A Concrete Approach to Abstract Algebra by W. W. Sawyer ($6 on Amazon!)

Though it goes a bit slow at times, it is by far the simplest, most intuitive book on Abstract Algebra in existence. Written for the non-mathematician, it does a great job of teaching the subject in simple, easy-to-understand prose. I couldn't put it down!

There are also two chapters on linear algebra, leading up to the final chapters, "vectors over fields" and "fields regarded as vector-spaces".

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Introduction to Analytic Number Theory - Tom M. Apostol.

When I bought this I really didn't want to put it down. It's a great book for exciting one's interest in the subject.

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